Trigonometry Lesson Objectives

Transcription

1 Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the Pythagorean theorem and trigonometric ratios to calculate side measures in right triangles. Angle Measures Convert between the decimal degree form and the degrees, minutes, seconds form of an angle measure. Use inverse trig functions to solve for angle measures in right triangles. Indirect Measure Solve word problems involving indirect measurement by using trigonometric ratios. Angles in the Coordinate Plane Identify coterminal angles. Determine in which quadrant the terminal side of an angle in standard position lies. The Unit Circle Determine the reference angle of a non-acute angle. Express a trig function of a non-acute angle in terms of an acute angle. Determine if a trig function value for a given angle will be positive or negative. Trigonometric Values of Special Angles Determine the exact trig function values for the 30, 45, and 60 angles. Use the 30, 45, and 60 angles as reference angles to determine trig values of other angles. Reciprocal Functions Express a reciprocal trig function of a non-acute angle in terms of an acute angle. Evaluate the reciprocal trig functions for the 30, 45, and 60 angles. Evaluate the reciprocal trig functions for angles having reference angles of 30, 45, and 60. Points on the Terminal Side Determine the trig values of quadrantal angles. Determine the trig function values of an angle in standard position given a point on the terminal side, or given one trig value and sign of another, or given one trig value and the quadrant in which the angle lies. Pythagorean Identities Evaluate trigonometric expressions. Simplify expressions by using trigonometric identities. Use the Pythagorean identities to determine the remaining trig function values of an angle in standard position when one trig value and sign of another, or one trig value and quadrant the angle lies in, are known.

2 Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY (CONT.) Radian Measure Convert between the degree and radian measure. Use the formula θ = s/r. Determine in which quadrant the terminal side of an angle in radian measure will lie. Reference Angles Determine the reference angle of an angle given in radian measure. Express a trig function of a non-acute angle in radian measure as a function of an acute angle. Evaluate trig functions of angles given in radian measure. Velocity Solve word problems about linear and angular velocity. Project: Parametric Equations Graph parametric equations defined by trigonometric functions. how to use right triangle trigonometry to solve problems. how to solve problems involving angular velocity. how to locate the quadrant in which an angle in standard position terminates. how to convert between various forms of angle measure. how to express trigonometric functions in terms of acute angles. how to simplify trigonometric expressions. how to determine values of trigonometric functions of an angle from a given point on the terminal side or from one trig function value.

3 Trigonometry Lesson Unit 2: GRAPHING AND INVERSE FUNCTIONS Graphing and Amplitude Identify the graphs of the sine, cosine, and tangent curves. Determine the amplitude of a sine or cosine function from a given equation or graph. State the domain and range of sine, cosine, and tangent curves of the form y = Asinx, y = Acosx, and y = Atanx. Project: The Reciprocal Functions Identify the domain and range of the reciprocal functions. Graph the reciprocal trigonometric functions. Period and Frequency Determine the frequency and period of a trigonometric function from the graph. Determine the frequency and period of a trigonometric function from the equation. Write the equation of a trigonometric function having a frequency other than 1. Vertical and Horizontal Translations Write the equation of a trig function with a given vertical or phase shift. Graph a trig function with a given vertical or phase shift. Determine the vertical and phase shifts of a trig function from a graph or equation. Sinusoidal Functions Graph, or identify the graph of, a sinusoidal function. Write an equation for a given sine or cosine graph. Determine the amplitude, period, phase shift, vertical shift, and range of sinusoidal functions. Inverse Functions Evaluate expressions containing inverse trig functions. Identify the graphs of the inverse trig functions. Inverse Reciprocal Functions Evaluate expressions containing inverse reciprocal trig functions. Identify the graphs of the inverse reciprocal trig functions. Trigonometric Equations: Part I Solve trigonometric equations using inverse functions. Solve trigonometric equations using factoring and the zero product property. Solve trigonometric equations using properties of equality. Trigonometric Equations: Part II Solve second-degree trigonometric equations using factoring and the zero product property. Solve trigonometric equations using the quadratic formula. Project: Modeling with Periodic Functions Write an equation of a sinusoidal function. Interpret the graph of a sinusoidal function. Graph a set of real-life data.

4 Trigonometry Lesson Unit 2: GRAPHING AND INVERSE FUNCTIONS (CONT.) how to solve trig equations. how to evaluate inverse trig functions. how to identify the amplitude, frequency, and phase shift of a trig function. how to graph trig functions using transformations. the graphs of the trig functions and their inverses. the domain and range for each inverse trig function.

5 Trigonometry Lesson Unit 3: ANALYTIC TRIGONOMETRY The Fundamental Trigonometric Identities Simplify trigonometric expressions using fundamental trigonometric identities. Use trig identities to find the remaining trig function values of an angle when one value is known. Determine equivalent trigonometric expressions using fundamental trigonometric identities. Proving Identities Solve trig equations using identities and substitution. Prove trigonometric identities. Cosine Addition Formula Evaluate trig functions using the cosine addition and subtraction formulas. Determine equivalent trigonometric expressions using the cosine addition and subtraction formulas. Sine Addition Formula Determine equivalent trigonometric expressions using the sine addition and subtraction formulas. Evaluate trig functions using the sine addition and subtraction formulas. Tangent Addition Formula Determine equivalent trigonometric expressions using the tangent addition and subtraction formulas. Evaluate trig functions using the tangent addition and subtraction formulas. Double-Angle Formulas Solve trig equations using identities and substitution. Determine equivalent trigonometric expressions using fundamental trigonometric identities. Evaluate trig functions of double-angle measures. Project: Solving Equations Graphically Solve trigonometric equations graphically. Half-Angle Formulas Determine equivalent trigonometric expressions using fundamental trigonometric identities. Evaluate trig functions of half-angle measures. Solve trig equations using identities and substitution. Converting Between Products and Sums Express a product of sine and cosine functions as a sum. Express a sum of sine and cosine functions as a product. Determine equivalent trigonometric expressions using product-to-sum and sum-to-product trigonometric identities. Project: Adding Waves Find the sum of two periodic waves. how to prove a trig identity. the trigonometric identities presented in this unit.

6 Trigonometry Lesson Unit 4: TRIGONOMETRIC APPLICATIONS Law of Sines Find missing angle and side measures of a triangle using the law of sines. Use the law of sines to solve indirect measure problems. Ambiguity and Area of a Triangle Find the area of a triangle using two sides and the included angle. Determine the number of triangles that can be formed given two sides and an angle. Law of Cosines: Finding a Side Find a side measure of a triangle using the law of cosines. Use the law of cosines to solve problems. Law of Cosines: Finding an Angle Find an angle measure of a triangle using the law of cosines. Use the law of cosines to solve problems. Project: Heron's Formula Prove Heron's formula using the law of cosines and algebra. Introduction to Vectors Find the direction of the resultant of two vectors. Add and subtract vectors. Find the magnitude of the resultant of two vectors. Define key terms associated with vectors. Vector Components Find the angles formed between two vectors and their resultant. Find the horizontal and vertical components of a vector. Add two position vectors given their terminal points. Navigation Application Solve word problems about navigation. Vector Multiplication Multiply a vector by a scalar. Find the dot product of vectors. Subtract vectors. how trigonometry of oblique triangles can be applied to real life situations. the law of cosines. how vectors are used to solve problems involving force. the law of sines. vectors and their components. the trig formula for finding the area of a triangle.

7 Trigonometry Lesson Unit 5: POLAR COORDINATES Introduction to Polar Coordinates Graph points in the polar plane. Convert between Cartesian and polar coordinates. Polar Equations Convert Cartesian equations to polar equations. Determine the slope of a line from its given polar equation. Convert polar equations to Cartesian equations. Project: Graphing in the Polar Plane Determine the center and radius of a circle from its given polar equation. Graph a polar equation using a table of values. Polar Curves Identify the graphs of polar curves rose, cardioid, and lemiçon. Recognize the equations of the polar curves rose, cardioid, and lemiçon. Polar Forms of Conics Write the polar equation of a conic section. Identify and graph a conic section from its equation in polar form. Determine the eccentricity of a conic section from its polar equation. Polar Form of Complex Numbers Represent complex numbers as points in the complex plane. Convert between the rectangular and polar forms of a complex number. Find the absolute value of a complex number. Multiply and Divide Complex Numbers Divide complex numbers in polar form. Multiply complex numbers in polar form. Powers and Nth Roots Find the nth roots of a complex number and use this to solve equations. Raise complex numbers in polar form to powers. Project: Fractals Apply iteration to complex numbers. Realize that new areas of mathematics continue to be explored and understood. Understand how computers have opened up new doors in mathematics. graphing in the polar plane plotting polar coordinates and polar curves. the relationship between the complex plane and the polar plane. finding products, quotients, powers, and roots of complex numbers in polar form. the relationship between the Cartesian plane and the polar plane. Unit 6: COURSE REVIEW AND EXAM major concepts presented in the trigonometry course in order to prepare for the final exam.